A 10.0-g marble slides to the left with a velocity of magnitude 0.400 m/s on the frictionless, horizontal surface of an icy New York sidewalk and has a head-on, elastic collision with a larger 30.0-g marble sliding to the right with a velocity of magnitude 0.200 m/s. Let +x be to the right.
a)Calculate the change in kinetic energy (that is, kinetic energy after the collision minus the kinetic energy before collision) for each marble. Compare the values you get for each marble.
Elastic collision assumes no energy lost. If energy is lost you need to know something about the situation after the collision.
To solve this problem, we'll use the principles of conservation of momentum and conservation of kinetic energy.
The formula for the change in kinetic energy is given by:
ΔKE = KE(after collision) - KE(before collision)
1. Let's start by calculating the initial kinetic energy (KE) for each marble.
For the left-moving marble:
Mass, m₁ = 10.0 g = 0.010 kg
Velocity, v₁ = 0.400 m/s
KE₁ = 0.5 * m₁ * v₁²
KE₁ = 0.5 * 0.010 kg * (0.400 m/s)²
KE₁ = 0.5 * 0.010 kg * 0.160 m²/s²
KE₁ = 0.008 J
For the right-moving marble:
Mass, m₂ = 30.0 g = 0.030 kg
Velocity, v₂ = -0.200 m/s (negative sign indicates rightward motion)
KE₂ = 0.5 * m₂ * v₂²
KE₂ = 0.5 * 0.030 kg * (-0.200 m/s)²
KE₂ = 0.5 * 0.030 kg * 0.040 m²/s²
KE₂ = 0.0006 J
2. Next, let's calculate the velocities of the marbles after the collision.
Since the collision is elastic, the total momentum before the collision is equal to the total momentum after the collision:
m₁ * v₁(before) + m₂ * v₂(before) = m₁ * v₁(after) + m₂ * v₂(after)
0.010 kg * 0.400 m/s + 0.030 kg * (-0.200 m/s) = 0.010 kg * v₁(after) + 0.030 kg * v₂(after)
0.004 kg m/s - 0.006 kg m/s = 0.010 kg * v₁(after) + 0.030 kg * v₂(after)
-0.002 kg m/s = 0.010 kg * v₁(after) + 0.030 kg * v₂(after)
3. Now, we can use the conservation of kinetic energy to find the velocities after the collision.
The sum of the kinetic energies after the collision should be equal to the sum of the kinetic energies before the collision.
KE₁(after) + KE₂(after) = KE₁(before) + KE₂(before)
0.5 * m₁ * v₁(after)² + 0.5 * m₂ * v₂(after)² = KE₁(before) + KE₂(before)
0.5 * 0.010 kg * v₁(after)² + 0.5 * 0.030 kg * v₂(after)² = 0.008 J + 0.0006 J
0.00005 kg * v₁(after)² + 0.00015 kg * v₂(after)² = 0.0086 J
-0.002 kg m/s = 0.010 kg * v₁(after) + 0.030 kg * v₂(after) (from step 2)
0.00005 kg * v₁(after)² + 0.00015 kg * v₂(after)² = 0.0086 J (from step 3)
4. We have two equations with two variables, v₁(after) and v₂(after).
Solving these equations, we find v₁(after) = 0.050 m/s and v₂(after) = -0.070 m/s.
5. Finally, let's calculate the final kinetic energies for each marble.
KE₁(after) = 0.5 * m₁ * v₁(after)²
KE₁(after) = 0.5 * 0.010 kg * (0.050 m/s)²
KE₁(after) = 0.000625 J
KE₂(after) = 0.5 * m₂ * v₂(after)²
KE₂(after) = 0.5 * 0.030 kg * (-0.070 m/s)²
KE₂(after) = 0.000735 J
Now, we can calculate the change in kinetic energy for each marble:
ΔKE₁ = KE₁(after) - KE₁(before)
ΔKE₁ = 0.000625 J - 0.008 J
ΔKE₁ = -0.007375 J
ΔKE₂ = KE₂(after) - KE₂(before)
ΔKE₂ = 0.000735 J - 0.0006 J
ΔKE₂ = 0.000135 J
Comparing the values, we find that the change in kinetic energy for the first marble (left-moving marble) is -0.007375 J, and for the second marble (right-moving marble) is 0.000135 J. The change in kinetic energy is larger for the first marble.
To calculate the change in kinetic energy for each marble, we need to know the initial and final velocities of each marble after the collision.
First, let's determine the final velocities of the marbles after the collision using conservation of momentum. We know that the total momentum before the collision is equal to the total momentum after the collision.
The initial momentum before the collision is calculated by multiplying the mass of each marble by their respective velocities:
Momentum before collision = (mass of first marble * velocity of first marble) + (mass of second marble * velocity of second marble)
Momentum before collision = (10.0 g * 0.400 m/s) + (30.0 g * (-0.200 m/s)) [please note that velocity of the second marble is negative as it is moving to the left]
Momentum before collision = 4.00 g*m/s + (-6.00 g*m/s)
Momentum before collision = -2.00 g*m/s
Since momentum is conserved, the momentum after the collision is also -2.00 g*m/s.
Now, since the collision is elastic (where kinetic energy is conserved), we can use conservation of kinetic energy to calculate the final kinetic energy for each marble.
The kinetic energy before the collision is calculated by:
Kinetic energy before collision = (1/2) * (mass of first marble) * (velocity of first marble)^2 + (1/2) * (mass of second marble) * (velocity of second marble)^2
Kinetic energy before collision = (1/2) * (10.0 g) * (0.400 m/s)^2 + (1/2) * (30.0 g) * (0.200 m/s)^2
Kinetic energy before collision = 0.40 g*m^2/s^2 + 0.20 g*m^2/s^2
Kinetic energy before collision = 0.60 g*m^2/s^2
Now, let's find the final kinetic energy for each marble. Since momentum is conserved, we can write the equation:
(mass of first marble * velocity of first marble) + (mass of second marble * velocity of second marble) = (mass of first marble * final velocity of first marble) + (mass of second marble * final velocity of second marble)
(10.0 g * 0.400 m/s) + (30.0 g * (-0.200 m/s)) = (10.0 g * final velocity of first marble) + (30.0 g * final velocity of second marble)
4.00 g*m/s - 6.00 g*m/s = 10.0 g * final velocity of first marble - 30.0 g * final velocity of second marble
-2.00 g*m/s = 10.0 g * final velocity of first marble - 30.0 g * final velocity of second marble
Since the collision is elastic, we also know that the relative velocities of the marbles remain unchanged.
final velocity of first marble = -initial velocity of second marble = -(-0.200 m/s) = 0.200 m/s
final velocity of second marble = -initial velocity of first marble = -(0.400 m/s) = -0.400 m/s
Now, let's calculate the final kinetic energy for each marble using the final velocities we found:
Kinetic energy after collision for first marble = (1/2) * (mass of first marble) * (final velocity of first marble)^2
Kinetic energy after collision for first marble = (1/2) * (10.0 g) * (0.200 m/s)^2
Kinetic energy after collision for second marble = (1/2) * (mass of second marble) * (final velocity of second marble)^2
Kinetic energy after collision for second marble = (1/2) * (30.0 g) * (-0.400 m/s)^2
Now, we can calculate the change in kinetic energy for each marble by subtracting the respective initial kinetic energy from the final kinetic energy:
Change in kinetic energy for first marble = Kinetic energy after collision for first marble - Kinetic energy before collision
Change in kinetic energy for second marble = Kinetic energy after collision for second marble - Kinetic energy before collision
Substituting in the calculations for the final and initial kinetic energies:
Change in kinetic energy for first marble = (1/2) * (10.0 g) * (0.200 m/s)^2 - (1/2) * (10.0 g) * (0.400 m/s)^2
Change in kinetic energy for second marble = (1/2) * (30.0 g) * (-0.400 m/s)^2 - (1/2) * (30.0 g) * (0.200 m/s)^2
Now, evaluate these expressions to get the final answers for the change in kinetic energy for each marble.